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We report the first electronic structure calculation performed on a quantum computer without exponentially costly precompilation. We use a programmable array of superconducting qubits to compute the energy surface of molecular hydrogen using two distinct quantum algorithms. First, we experimentally execute the unitary coupled cluster method using the variational quantum eigensolver. Our efficient implementation predicts the correct dissociation energy to within chemical accuracy of the numerically exact result. Second, we experimentally demonstrate the canonical quantum algorithm for chemistry, which consists of Trotterization and quantum phase estimation. We compare the experimental performance of these approaches to show clear evidence that the variational quantum eigensolver is robust to certain errors. This error tolerance inspires hope that variational quantum simulations of classically intractable molecules may be viable in the near future.Comment: 13 pages, 7 figures. This revision is to correct an error in the coefficients of identity in Table

The superposition principle is a fundamental tenet of quantum mechanics. It allows a quantum system to be 'in two places at the same time', because the quantum state of a physical system can simultaneously include measurably different physical states. The preparation and use of such superposed states forms the basis of quantum computation and simulation. The creation of complex superpositions in harmonic systems (such as the motional state of trapped ions, microwave resonators or optical cavities) has presented a significant challenge because it cannot be achieved with classical control signals. Here we demonstrate the preparation and measurement of arbitrary quantum states in an electromagnetic resonator, superposing states with different numbers of photons in a completely controlled and deterministic manner. We synthesize the states using a superconducting phase qubit to phase-coherently pump photons into the resonator, making use of an algorithm that generalizes a previously demonstrated method of generating photon number (Fock) states in a resonator. We completely characterize the resonator quantum state using Wigner tomography, which is equivalent to measuring the resonator's full density matrix.

Nature 454, 310 (2008) Recommended and Commentary by Steven M. Girvin, Yale University Microwaves, despite their name, are particles. However the photon quanta of microwave fields are rather pusillanimous. They carry four to five orders of magnitude less energy than optical photons and are correspondingly vastly more difficult to detect and count. Nevertheless, recent progress in atomic cavity QED [1] and superconducting circuit QED [2] has achieved this. Single-photons-on-demand as well as coherent superpositions of 0 and 1 photons have been generated in a microwave resonator electrical circuit.[3]A classical signal generator produces a sine wave of constant amplitude, frequency and phase. The quantum equivalent (produced by a laser or a microwave signal generator) is a so-called coherent state. Because the phase is sharply defined, the photon number (which is the conjugate variable), is necessarily ill-defined. The number of photons to be found in a coherent pulse is in fact Poisson distributed. As a result, a coherent pulse which contain N photons on average will have a variance in photon number of √N. These closest cousins to classical waves are of course useful but not terribly exciting. There is great current interest in generating highly non-classical states of the electromagnetic field for purposes of quantum communication and quantum information processing. One interesting and highly non-classical class of states are the Fock states. These are electromagnetic pulses which contain exactly n photons where n is some specified integer. Because they have definite photon number, the phase suffers complete quantum uncertainty. Hence the electric field of such pulses is completely uncertain, a fact which has recently been verified. [3] Hofheinz et al. have made a tour-de-force advance by deterministically generating photon number Fock states containing up to N = 6 photons (N = 15 in recent unpublished work) using a superconducting qubit coupled to a resonator.The resonator supports discrete modes at integer multiples of the fundamental. Because the modes are widely spaced in frequency for short res-1